Who are these of Matthew 8?

Because not only did they not hearken to that Prophet like unto Moses, they slew Him.

On what basis do you define 'cast out into the outer darkness; in that place there will be weeping and gnashing of teeth' as the lake of fire?

On what basis have you determined these to be unregenerate?

Where did you get the idea these were cast into the lake of fire?

They say unto him, He will miserably destroy those miserable men, and will let out the vineyard unto other husbandmen, who shall render him the fruits in their seasons. Jesus saith unto them, Did ye never read in the scriptures, The stone which the builders rejected, The same was made the head of the corner; This was from the Lord, And it is marvelous in our eyes? Therefore say I unto you, The kingdom of God shall be taken away from you, and shall be given to a nation bringing forth the fruits thereof....And when the chief priests and the Pharisees heard his parables, they perceived that he spake of them. Mt 21.41-45

Noah foretold of it here:

God enlarge Japheth, And let him dwell in the tents of Shem.......Gen 9:27

Paul wrote of it here:

But if some of the branches were broken off, and thou, being a wild olive, wast grafted in... Ro 1:17

They are called the 'sons of the kingdom' because:

4 who are Israelites; whose is the adoption, and the glory, and the covenants, and the giving of the law, and the service of God, and the promises;
5 whose are the fathers, and of whom is Christ as concerning the flesh, who is over all, God blessed for ever. Amen. Ro 9

 

I think that Isaiah 60 could be telling us of the same "outer darkness" that Christ speaks of:

NIV
1 “Arise, shine, for your light has come,
and the glory of the Lord rises upon you.
2 See, darkness covers the earth
and thick darkness is over the peoples,
but the Lord rises upon you
and his glory appears over you.
3 Nations will come to your light,
and kings to the brightness of your dawn.​

I think that this is speaking of the millennium.

Either that, or the "outer darkness" is speaking of Sheol/Hades (Job 10:21-22). (Or maybe even both!).

As for the "children of the kingdom", this is speaking of the Jews.

 

Fire is representative of God's wrath both in the temporal and eternal realms:

http://www.baptistboard.com/showthread.php?p=1940283#post1940283

...furnace of fire synonymous with lake of fire? I don't think so.

How much soever she glorified herself, and waxed wanton, so much give her of torment and mourning: for she saith in her heart, I sit a queen, and am no widow, and shall in no wise see mourning. Therefore in one day shall her plagues come, death, and mourning, and famine; and she shall be utterly burned with fire; for strong is the Lord God who judged her. And the kings of the earth, who committed fornication and lived wantonly with her, shall weep and wail over her, when they look upon the smoke of her burning, standing afar off for the fear of her torment, saying, Woe, woe, the great city, Babylon, the strong city! for in one hour is thy judgment come. And the merchants of the earth weep and mourn over her, for no man buyeth their merchandise any more; ………The merchants of these things, who were made rich by her, shall stand afar off for the fear of her torment, weeping and mourning; saying, Woe, woe, the great city, she that was arrayed in fine linen and purple and scarlet, and decked with gold and precious stone and pearl! for in an hour so great riches is made desolate. And every shipmaster, and every one that saileth any wither, and mariners, and as many as gain their living by sea, stood afar off, and cried out as they looked upon the smoke of her burning, saying, What city is like the great city? And they cast dust on their heads, and cried, weeping and mourning, saying, Woe, woe, the great city, wherein all that had their ships in the sea were made rich by reason of her costliness! for in one hour is she made desolate. Rev 18.7-11,15-19

 

Then you should be able to take this great amount of knowledge that you've amassed from Bible Gateway and answer your own questions. Outer darkness MUST equate to the lake of fire/second death/eternal punishment. Absolutely must.

 

11 And I say unto you, that many shall come from the east and the west, and shall sit down with Abraham, and Isaac, and Jacob, in the kingdom of heaven: Mt 8

5 even when we were dead through our trespasses, made us alive together with Christ (by grace have ye been saved),
6 and raised us up with him, and made us to sit with him in the heavenly places, in Christ Jesus: Eph 2

When's the last time that you've sit with Christ in the heavenly places?